Lecture for Week 9 2013 Spring. Numbers can be represented in many ways. We are familiar with the decimal system since it is most widely used in everyday

Lecture for Week 9Lecture for Week 9

2013 Spring 2013 Spring

Numbers can be represented in many ways. We are familiar with the decimal system since it is most widely used in everyday life. The decimal system is an example of a positional number system. Each digit has a place value that depends on its position with respect to the decimal point.

The Binary Number System

Example:2386.75 = 2*103+3*102+8*10+6*100+7*10-1+5*10-2

where 6 represents number of units, 8 number of tens, 3 number of hundreds, 2 number of thousands

To the right of decimal point we have7 number of tenths,5 number of hundredths,and so on.


The decimal system is said to use a base of 10 because the place values are powers of 10. Because we use the decimal system all the time it is easy to forget that there is no mathematical reason for using powers of 10 as the place values. The choice of 10 probably arose because people used their 10 fingers for counting.But in fact we can use any number as the base of a positional number system.A familiar example of non-decimal number system is the subdivision of an hour into 60mins, a minute into 60sec.2 h 26 m 35 s = 2*602 + 26*601 + 35*600


It turns out that in Computing bases 2, 8 and 16 are particularly useful.Base 2 is the binary number systemBase 8 is the octal number systemBase 16 is the hexadecimal number systemThe binary number system is a positional number system that uses 2 as the base.For the purpose of this course we shall be working only with non-negative integers ( Z+{0} ).


Example:101102 = 1*24 +0*23 +1*22 +1*21 +0*20

= 16 + 0 + 4 + 2 +0 = 22

Note that subscript 2 represents base 2 and it is necessary to state it unless we work in the decimal number system.

(Expressing negative numbers or real numbers is not really any different)


The table shows the integers from 0 to 20 in their binary representation


Binary Decimal Binary Decimal01

1011

100101110111

100010011010

0123456789

10

10111100110111101111

1000010001100101001110100

11121314151617181920

The binary system uses the 2 binary digits 0 and 1. Every number in binary will appear as strings of zeros and ones.

In octal as strings of 0,1,2,…7In decimal as strings of 0,1,2,…,9 In hexadecimal as strings of 0,1,2,…,9,A,B,C,D,E,F where A represents 10, B represents 11, C represents 12, D represents 13, E represents 14 and F represents 15.


Base-10 number system Consist of ten numbers- 0 to 9 Common type of number system used everyday.

Examples- (20)10 , (50)10 , (145)10 , etc.

Decimal number system

From the table we can see that1. odd numbers end with 12. even numbers end with 0

These two lines can be summarised as follows:

The last digit in the binary representation is the remainder after dividing the number by 2.

Conversion from Decimal to Binary

So the first step is to divide (integer division) n by 2 to obtain quotient and remainder (both whole numbers and remainder can only be 0 or 1).

Quotient remainder form

PASCAL notation:

N DIV 2 = quotient (7 DIV 2 = 3)N MOD 2 = remainder. (7 MOD 2 = 1)


In order to proceed further we make another observation

By removing the rightmost digit in a binary representation we obtain the binary

representation of n DIV 2.


Example1:14 = 11102

removing last digit (0) we obtain 111 which in base 10 represents 7.14 DIV 2 = 7

Example211 = 10112

removing last digit (1) we obtain 101 which in base 10 represents 5.11 DIV 2 = 5 ( DIV only returns quotient)


Each time we perform division we record remainder until n DIV 2 (quotient) becomes zero. Then we return the string of remainders in reverse order.


Convert (250)10 to binary

250/2=125 remainder is 0 125/2=62 remainder is 1 62/2=31 remainder is 0 31/2=15 remainder is 1 15/2=7 remainder is 1 7/2=3 remainder is 1 3/2=1 remainder is 1 1/2=0 remainder is 1 Therefore….(250)10=(11111010)2

Decimal to binary Conversion

Convert 25 into binary.

25 = 12*2+112 = 6*2+06 = 3*2+03 = 1*2+11 = 0*2+1Therefore, 25 = (11001)2

Decimal to binary Conversion

Taking binary number as (11111010)2 0×20 =0 1 ×21 =2 0 ×22 =0 1 ×23 =8 1 ×24 =16 1 ×25 =32 1 ×26 =64 1 ×27 =128 Adding the numbers we get…0+2+0+8+16+32+64+128=250 Therefore….(11111010)2 =(250)10

Binary to decimal Conversion

Convert the following as indicated

1.34 to binary 2.64 to binary 3.123 to binary 4.111000 to decimal 5.10101010 to decimal 6.101110011 to decimal 7.878 to binary

Practice Problem

Adding binary numbers is a very simple task, and very similar to the longhand addition of decimal numbers. As with decimal numbers, you start by adding the bits (digits) one column, or place weight, at a time, from right to left. Unlike decimal addition, there is little to memorize in the way of rules for the addition of binary bits:

Addition and Multiplication in Binary

0 + 0 = 0 1 + 0 = 1 0 + 1 = 1 1 + 1 = 10 1 + 1 + 1 = 11

Just as with decimal addition, when the sum in one column is a two-bit (two-digit) number, the least significant figure is written as part of the total sum and the most significant figure is "carried" to the next left column. Consider the following examples:


The addition problem on the left did not require any bits to be carried, since the sum of bits in each column was either 1 or 0, not 10 or 11. In the other two problems, there definitely were bits to be carried, but the process of addition is still quite simple.


Binary multiplication is actually much simpler than decimal multiplication. In the case of decimal multiplication, we need to remember 3 x 9 = 27, 7 x 8 = 56, and so on. In binary multiplication, we only need to remember the following,

Binary multiplication

0 x 0 = 00 x 1 = 01 x 0 = 01 x 1 = 1

Note that since binary operates in base 2, the multiplication rules we need to remember are those that involve 0 and 1 only.

101 x11First we multiply 101 by 1, which produces 101. Then we put a 0 as a placeholder as we would in decimal multiplication, and multiply 101 by 1, which produces 101. 101 x11 101 1010 <-- the 0 here is the placeholder


The next step, as with decimal multiplication, is to add. The results from our previous step indicates that we must add 101 and 1010, the sum of which is 1111.

101 x11 101 1010 1111



Multiply 11010 and 1011

Multiply the following:1.1011 x 112.110 x 1013.111 x 104.1110 x 11015.10110 x 101

Practice Problem

Subtraction ( and division) of binary numbers is performed in the same way as with decimal numbers. Binary subtraction is simplified as well, as long as we remember how subtraction and the base 2 number system. Let's first look at an easy example.

Subtraction and Division

Since addition is opposite operation of subtraction the table for subtraction can be deduced from the addition table. 0 + 0 = 0 then 0 - 0 = 0;1 + 0 = 1 then 1 - 0 = 1;0 + 1 = 1 then 1 - 1 = 0;1 + 1 = 0 (and 'carry' 1)then 0 - 1 = 1 ( and 'borrow' 1 from the next position ).


Note that the difference is the same if this was decimal subtraction. Also similar to decimal subtraction is the concept of "borrowing." Watch as "borrowing" occurs when a larger digit, say 8, is subtracted from a smaller digit, say 5, as shown below in decimal subtraction.


For 10 minus 1, 1 is borrowed from the "tens" column for use in the "ones" column, leaving the "tens" column with only 2. The following examples show "borrowing" in binary subtraction.


Binary division is almost as easy, and involves our knowledge of binary multiplication. Take for example the division of 1011 into 11.


To check our answer, we first multiply our divisor 11 by our quotient 11. Then we add its' product to the remainder 10, and compare it to our dividend of 1011.

The sum is equal to our initial dividend, therefore our solution is correct.


Complements are used for storing numbers in the computer as well as for subtraction. In subtraction instead of calculating A - B we can add A to 1-complement of B plus 1 or by adding 2-complement of B to A.Unfortunately, this method needs a little bit more refining. Before we continue with it we first define complements of a binary number.

One's and Two's Complements

One's complement of a binary number A is a binary number obtained by subtracting each digit of A from 1.

Alternatively, one's complement of a binary number A is a binary number such that:Each digit of A is changed into its opposite (i.e. 1 into 0, 0 into 1)


Example 1's complement of 110101 is 001010Two's complement of a binary number A is a binary number obtained by adding 1 to one's complement of A.

ExampleFrom the above exampleA is 110101 Then 1's compl. is 0010102's compl. is 001010 + 1 ----------- 001011


Any questions

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Lecture for Week 9 2013 Spring. Numbers can be represented in many ways. We are familiar with the decimal system since it is most widely used in everyday